*Hi Holly,*

*I'm a chemistry major, and I thought I should be able to answer this question, but I'm not sure how to go about it. *

*I take a form of vitamin C which, according to the label, is “magnesium ascorbate”. Wouldn’t this also contain some magnesium?*

*Is there a way to calculate how much magnesium each tablet contains?*

*The label says 2 tablets contain*

*Vitamin C 500 mg as magnesium ascorbate and ascorbic acid*

*Thanks!*

*Haley*

Hi Haley,

Your question is such a fun calculation that I can’t resist giving it a whirl. If you have studied stoichiometry, this uses the same tools, although there is no reaction involved.

The question demonstrates the power of **unit conversion**, which, once mastered, can unlock information in many fields, not just in chemistry, and not just in science. I love giving this powerful tool away.

So if you don’t mind, I’ll diverge a bit into ** why unit conversion works. So often students are told how to do unit conversion, but not why it works. **You can skip the theory part if you want, of course.

**First, I have some bad news about your question. **

Because your vitamin C comes from a mixture (*magnesium ascorbate and ascorbic acid*), we will not be able to tell exactly how much magnesium is in each tablet unless we know the percents of each in that mixture. I don’t have that information.

But we can certainly say the *maximum* amount of magnesium in each tablet, if we make the wrong assumption that 100 % of the vitamin C is *all* in the form of magnesium ascorbate first.

This is not true, obviously, but it gives us an upper limit of X, and we can then say with confidence that the magnesium in each tablet is less than X, but not more than X. It will give us a range.

Magnesium ascorbate tastes bitter, by the way. I'm guessing the makers of these tablets are using it as a buffer for the more tangy tasting ascorbic acid, which will taste sour.

__First I will digress into unit conversions. (Skip to the bottom if you want the answer)__

Forgive me for doing a very simple problem. My goal is not to show you how to convert feet to inches. My goal is explain *why* the math works so it can be applied with confidence to more complex problems.

What’s a unit?** A unit is a descriptor for a number.** We have units for all sorts of parameters, like length, mass, loudness, etc, like kilometers, grams, decibels. You can convert one unit into another, in some cases, if they are the same dimension, or relate numbers of things.

**I find a unit-less number troubling.** I counsel my students that if they are **writing down a number without a unit, something could be wrong**. *Usually* numbers require units in science.

You can’t just say *“I am five” *without meaning being lost. Are you five *years* old? Five *feet* tall? Five *lightyears* away from us in space? If the context is not clear, the patient can die.

Some dimensions are naturally unit-less, but most are not. So, when you consider a number, always consider what unit it is married to.

**How to use conversion factors and WHY THEY WORK (math theory)**

Here’s a standard, simple unit conversion question:

I am (barely) 5 feet tall. how many inches is this?

This method is also called *dimensional analysis:*

**Step 1.** **make a list of the givens i**n the problem. What is a

*given*?

A given is a number tied (next to) to a unit. **Do not separate the number from the unit! Why?**

Just as 5a means* 5 times a* in algebra, 5 feet means *5 times 1 feet*. (the 1 is understood)

What this means is that **you can't separate the 5 from the a. You would destroy the meaning of the phrase if you do this.**

I put that in bold because separating a number from its unit is the root of so many unit conversion problems going badly. Only a proper unit conversion can change the unit from one to another.

So, we write down, 5 ft.

There may be many givens in a problem, if so, write them all down in a list.

**2. Find the wanted.** What is the

*wanted*?

The wanted is usually* a unit or a dimension with no number associated with it.* It’s what you are being asked to calculate.

For example, "inches." This is the unit you want for your answer. **When you get a number next to this unit, you know you are done!**

**3. Find a conversion factor that relates the unit of the given to the unit of the wanted.**

**A conversion factor is a statement of truth that relates one unit to another. **

A conversion factor can always be stated in the form

**number unit = number unit**

for example,

12 inches = 1 foot

to derive conversion factors from metric modifiers, you substitute the meaning of the metric modifier symbol for a number, for example,

In a table of metric modifiers, you see that "*k" means 1,000. *

So substitute the k for the number 1000

**k**m means **1000** m, in other words

1 **k**m = **10000** m (all you have done is substituted 1000 for k since they mean the same thing)

**Step 4. Your conversion factor is equivalent to the number 1. (thunderclap here!) **

**There are powerful things that the number one can do. You can multiply by one and get the same thing back.**

Realize that *if a = b,* you can put a over b, or b over a, and still have the number one.

*if a = b,*

a/b =1 and

b/a = 1

Thus, for your conversion factor, you can say that

1 feet/12 inches is equivalent mathematically to the number 1.

Also,

12 inches/1 foot is equivalent mathematically to the number 1.

Both **1 feet/12 inches** and **12 inches/1 feet **are conversion factors.

Thus, **another form of conversion factor looks like a ratio:**

**number unit/number unit **

**providing** that the term on the top and the term on the bottom are equal to each other!

Conversion factors are **always equivalent to the number 1,** because *when you divide the same thing by itself you get the same number back.*

**Another wonderful thing about the number 1 is that you can multiply it by anything and get the same thing back!**

You will always have the possibility of *two* ratios (since you can have either a/b or b/a) and for any given problem **BUT** **only one of these two ratios will give you the correct answer.**

In order to see which one gives you the correct answer,** ALWAYS use units in your calculations! **ALWAYS!

The** units behave just like algebraic variables and will tell you whether you are doing it right or not**:

**Step 5. Multiply the given by the correct conversion factor ratio. **If you use the wrong ratio, you will not see the unit of the wanted for your answer:

First, let’s just try the *wrong* ratio to see what happens:

5 feet x 1 feet/12 inches = 0.4 **feet**^{2}**/inches** ??????

notice the **units of this answer make no sense, so it can't be the right **conversion factor ratio.

Second try:

5 feet x 12 inches/1 feet = 60 **inches**

notice both "feet" divide and leave only the unit "inches" which is what you want, the "wanted", thus you know you are done.

__________

**End of unit conversion theory! Now let’s use this on the vitamin C question. **

Step 1. What are the *givens*?

500 *mg ascorbate*

2 *tablets*

The words *mg ascorbate* and *tablets* are both units. *Tablets* is a funny unit, but it is still a unit. A unit can be a number of items, like *a dozen* of donuts.

Step 2. What is the *wanted*? Haley wanted to know *how much magnesium* was in two tablets. She didn’t specify a unit, but a useful unit would be one we already use for recommended daily allowances, like mg.

So, let’s solve for **mg of magnesium**. (the *wanted*)

3. Find a conversion factor that links the given to the wanted. Sometimes this requires finding more than one conversion factor. They are all equivalent to the number one though!

**You can’t convert the mass of one compound directly into the mass of another compound.** You could think of the identity of the compound as being part of the unit. You have to first convert to the number of items.

This is as if you are making a ton of cheese sandwiches, with every sandwich having exactly TWO slices of bread for every ONE slice of cheese, and you are given the *weight* of cheese only, and you have to calculate the number of slices of bread. You will have to convert the weight of cheese to a number of slices before you can relate cheese slices to bread.

We can relate the number of magnesiums to the number of ascorbates, first:

Here’s where I will have to get chemical. *Magnesium ascorbate* is not the same thing as *magnesium*, and and neither of these is the same as ascorbate (vitamin C). We have to account for this by looking at individual numbers of things.

Chemists like to use batches of things, since items are small. The most convenient number of things is the mole. Just as people talk about dozens of donuts, and we know this means batches of twelve, moles is batches of 6.02 X 10 ^{23} things. It’s a lot of things, but we need that since we are dealing with very small things in large numbers. (Why that number—Avogadro’s number— is so special is the subject of another post!)

One of the wonderful things about moles is that you can look up the **molar mass **of any chemical (these days, online, but you can calculate a molar mass given the formula if you have the periodic table). **The molar mass is simply how much one mole of a substance weighs in grams.**

The molar mass is a wonderfully useful conversion factor. It is always given with the same units: **moles per gram**. (You might see the unit given as *Daltons*, in honor of a famous chemist, but it’s the same thing as moles per gram. I make my students memorize that molar mass comes in units of moles per gram, because it is such a useful conversion factor.)

I just looked up the molar mass of ascorbate. (You can actually calculate its molar mass on your own with a periodic table, since all periodic tables give molar masses for all of the elements. You just use the formula of ascorbate and add up all the individual molar masses of all the elements in the formula, noting that you have more than one of certain element in the formula. The formula for ascorbate is C_{6}H_{7}O_{6})

**The molar mass of ascorbate is 175 grams per mole.** (Ascorbic *acid* is another form of vitamin C that has only one more hydrogen, increasing its weight to 176 g/mole, and for our calculations, the difference is not significant.)

So we have the conversion factor:

**175 grams of ascorbate = 1 mole of ascorbate**

But we are not given grams. We are given milligrams (mg). We need at least one more conversion to relate what we are given (mg) to the conversion factor which uses a different mass unit (grams).

*Milli* means a *thousandth*, so we can say that

**m**g = **0.001** gram using simple substitution of *0.001* with *m*.

This is yet another conversion factor.

Let’s see how far we can get with these.

I want to convert mg to grams so I can relate it to the molar mass conversion unit:

500 **mg** ascorbate X 0.001 grams/1 **mg** = 0.500 grams ascorbate

(notice how the boldfaced units canceled leaving only the other units behind)

Now we can relate grams of ascorbate to moles of ascorbate using its molar mass. *Whenever you want to convert from moles to grams, or from grams to moles, you need to use the molar mass. *

0.500 **grams ascorbate** X 1 mole ascorbate/175 **grams ascorbate **= 0.002857…moles ascorbate.

Or, 2.86 X 10^{3} moles ascorbate if you like to use scientific notation.

Notice how I turned the molar mass “upside down” on the right so the grams of ascorbate would cancel.

Now we are going to assume all this ascorbate is in the form of magnesium ascorbate. This will give us an upper maximum for the amount of magnesium there is in two tablets.

*Magnesium ascorbate* is an ionic compound. Ionic compounds are made of positively and negatively charged ions stuck together in a crystal lattice, almost all are solid at room temperature. I wouldn’t expect the non chemist to know this, but ascorbate has a charge of negative one. Magnesium ion has a charge of plus two.

A quick look up of the formula confirms to me that **there are TWO ascorbates for every magnesium**. (picture)

THIS is a conversion factor:

**1 magnesium always comes with (or is equivalent to) 2 Ascorbates (in magnesium ascorbate) **

**so, 1 magnesium = 2 Ascorbates (when in the form of magnesium ascorbate)**

Moles is a *number of things*, and we have work with this unit to relate the number of ascorbates to the number of magnesium ions.

We need to change moles of ascorbate to moles of magnesium. If moles bothers you as a unit, just think of it as dozens, only much bigger.

**For every two units of ascorbate, there is one magnesium. **

So, if we had *two* *dozen* ascorbates, it would make sense that we had *one dozen* magnesiums, right?

Therefore if we had 2 moles of ascorbate, we would also have 1 moles of magnesium.

This gives us another conversion factor:

**2 moles ascorbate = 1 mole magnesium (within magnesium ascorbate)**

This type of conversion factor comes from the formula for a compound. (This is often done in stoichiometry for those of you who are chemistry students.) You can relate numbers of things in a formula, which expresses a truth about that compound. Just make sure you are talking about the right compound!

0.002857 moles ascorbate X 1 mole magnesium/2 moles ascorbate = 0.0014285….moles of magnesium.

Yes, all we did was divide by two, but it changed the *units*. Very important.

Now we are cooking with gas, baby! Almost there!

Anytime you want to go from numbers of things to grams, or from grams to numbers of things, you can use the molar mass. (I actually make my students memorize this because it is so useful.)

So let’s look up the molar mass of magnesium. It’s an element, so it will be in the periodic table.

The molar mass of magnesium is 24.3 grams per mole.

0.0014285 moles magnesium X 24.3 grams magnesium/1 mole magnesium = 0.0347 grams of magnesium

or, converting to mg

0.0347 grams of magnesium X 1 mg/0.001 grams = 34.7 mg of magnesium per two tablets (MAX!)

That’s not a lot, and this is based on the wrong assumption that all the ascorbate comes from magnesium ascorbate.

**So, two tablets probably contain less than 35 mg of magnesium. Tadaa!**

This is not a lot, given that the RDA for magnesium is around 400 mg (420 mg/day for men, 320 mg/ day for women.)

Wow thanks for this!

I didn’t know I could make a wrong assumption and then calculate the maximum possible mass from that. Cool.

I like the number theory too. I never thought about conversion factors being like the number 1 but it explains why they work.

I’m sharing that with the high school class I tutor if that’s okay?

Cheers

Posted by: Haley Anderson | 07/10/2024 at 10:06 AM